Applying to the penalty-constrained least-square misfit (REF ) a Lagrange
multiplier approach (see [1]}), we can define an \(\varepsilon \) -dependent Lagrangian
\((u,v)\mapsto \mathcal {L}^\varepsilon : V(\Omega _t)^2\mapsto \mathbb {R}\)  as
\(\mathcal {L}^\varepsilon (u, v; \Omega _t) =\mathcal {J}(u; \Omega _t)-\langle \partial ^\varepsilon _u \mathcal {E}(u; \Omega _t), v \rangle .\) 
